On the existence problem for a double mean field equation and its application to prescribed curvature problems
Abstract
Let (M,g) be a compact Riemann surface with smooth boundary ∂ M. Let K(x)>0, h(x)> 0 or h(x)≡ 0 be two smooth functions on M and ∂ M respectively. In this paper, we consider the following double mean field type equation with a nonlinear Neumann boundary condition related to a prescribed curvature problem labelequation:01 −Δg u+4π/|M| =2ρKeu/(∫MKeu dvg), & in M, [1.2em] ∂νu =2ρ'heu/2/(∫∂ Mheu/2 dsg), & on ∂ M, where ρ> 0, ρ'≥ 0 and ρ+ρ'=2π, ν is the outward unit normal vector to ∂ M. When h≡ 0, the second equation changes to ∂νu=0 on ∂ M. By introducing a boundary integration constant, we obtain a sufficient condition under which \eqref{equation:01} admits a solution. As a geometric application, we solve the prescribed curvature equation for both the Gauss curvature and the geodesic curvature in the case that M is topologically a disk. When (M,g) is the standard hemisphere 𝕊2+, let p∈ ∂ 𝕊2+ be a maximum point of Φ(q)=h(q)+√(h(q)2+K(q)), q∈∂ 𝕊2+. Suppose Kν(p)+2Φ(p)(−Δ∂ 𝕊2+)1/2h(p)<0. Then the prescribed curvature problem −Δg u+4π/|M|=2Keu, in 𝕊2+, ∂νu=2heu/2, on ∂ 𝕊2+. admits a smooth solution.
Related
- Existence of disks with prescribed mean curvature and contact angle in arbitrary codimension
- Interior curvature estimates for convex solutions of a prescribed curvature quotient equation
- Curvature Estimates and Prescribed Curvature Problems for Sum Hessian Equations on Star-Shaped Hypersurfaces
- Gauge rigidity in an inverse problem for the prescribed Gaussian curvature equation
- Liouville Rigidity and Universal Spacelikeness Estimates for a Lorentzian Prescribed Mean Curvature Equation
BibTeX
@misc{li2026existence,
title = {{On the existence problem for a double mean field equation and its application to prescribed curvature problems}},
author = {Jiayu Li and Lei Liu and Yanjun Liu and Mingyuan Zhu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17571},
eprint = {2610.17571},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17571},
note = {CC0},
}
RIS
TY - UNPB
TI - On the existence problem for a double mean field equation and its application to prescribed curvature problems
AU - Jiayu Li
AU - Lei Liu
AU - Yanjun Liu
AU - Mingyuan Zhu
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17571
UR - https://arxiv.org/abs/2610.17571
AB - Let (M,g) be a compact Riemann surface with smooth boundary ∂ M. Let K(x)>0, h(x)> 0 or h(x)≡ 0 be two smooth functions on M and ∂ M respectively. In this paper, we consider the following double mean field type equation with a nonlinear Neumann boundary condition related to a prescribed curvature problem labelequation:01 −Δ_g u+4π/|M| =2ρKe^u/(∫_MKe^u dv_g), & in M, [1.2em] ∂_νu =2ρ'he^u/2/(∫_∂ Mhe^u/2 ds_g), & on ∂ M, where ρ> 0, ρ'≥ 0 and ρ+ρ'=2π, ν is the outward unit normal vector to ∂ M. When h≡ 0, the second equation changes to ∂_νu=0 on ∂ M. By introducing a boundary integration constant, we obtain a sufficient condition under which \eqref{equation:01} admits a solution. As a geometric application, we solve the prescribed curvature equation for both the Gauss curvature and the geodesic curvature in the case that M is topologically a disk. When (M,g) is the standard hemisphere 𝕊^2_+, let p∈ ∂ 𝕊^2_+ be a maximum point of Φ(q)=h(q)+√(h(q)^2+K(q)), q∈∂ 𝕊^2_+. Suppose K_ν(p)+2Φ(p)(−Δ_∂ 𝕊^2_+)^1/2h(p)<0. Then the prescribed curvature problem −Δ_g u+4π/|M|=2Ke^u, in 𝕊^2_+, ∂_νu=2he^u/2, on ∂ 𝕊^2_+. admits a smooth solution.
ER -